The main aim of this course is the study of basic special functions and proves the properties and relations related to these functions. Furthermore, the simple sets of polynomials are discussed.

Learning Outcomes

Learn basic concepts of special functions.
Define gamma, beta and hypergeometric functions and prove some properties involving these functions.
Discuss some applications of the gamma, beta function and hypergeometric functions,
Define simple sets of polynomials.
Define normalization, Bessel’s inequality, generating functions, differential equations, recurrence relations.

The Weierstrass gamma function, Euler integral representation of gamma function.
Relations satisfied by gamma function.
Euler’s constant, The order symbols o and O, properties of gamma function.
Beta function, integral representation of beta function.
Relation between gamma and beta functions, properties of beta function.
Legendre’s duplication formula.
Gauss’ multiplication theorem.
Hypergeometric series, the functions F(a,b;c;z) and F(a,b;c;I), integral representation of hypergeometric function.
The hypergeometric differential equation, The contiguous relations, Simple transformations.
A theorem due to Kummer.
Confluent hypergeometric series, Integral representation of confluent hypergeometric function, the confluent hypergeometric.
Differential equation, Kummer’s first formula.
Simple sets of polynomials, Orthogonality.
The three term recurrence relation, The Christofell-Darboux formula.
Normalization, Bessel’s inequality, Generating functions, Differential equations, Recurrence relations.